Question

find the inverse of the matrix if it exists p=[4,4 4,5]

Answer 1

\[\begin{vmatrix}
4 & 4\\
4 & 5
\end{vmatrix}=20-16=4\neq0.\]It has an inverse.

Answer 2

The inverse of a 2x2 matrix
\[\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\]
is \[\left[\begin{matrix}d & -b \\ -c & a\end{matrix}\right]\]
divided by the determinant of the matrix (ac-bd).

Answer 3

Sorry, the determinant should read (ad-bc).

Answer 4

Given \(A\) is a \(2\times2\) matrix, it's inverse can be computed by:
\[A^{-1}=\frac{1}{ad-cb}\begin{bmatrix}
d & -b\\
-c & a
\end{bmatrix}.\]Do you know how to do that, nickyd63?

Answer 5

no that's what i need help with
i just do not understand it

Answer 6

All you have to do is plug the matrix values into the above expression, like this:\[A^{-1}=\frac{1}{4\cdot5-4\cdot4}\begin{bmatrix}
5 & -4\\
-4 & 4
\end{bmatrix}.\]Can you simplify that?

Answer 7

No, im still trying to figure out how you did that

Answer 8

From Across's matrix, you'll see that the top left element a has exchanged places with element d in the inverse, and the elements b and c have changed sign. That's how she got the inverse.
\[(1/4)\left[\begin{matrix}5 & -4 \\ -4 & 4\end{matrix}\right]\]
However, you still have to divide each element by the determinant, calculated using ad-bc, or in the given case, (4)(5)-(4)(4)=4.
The final result becomes:
\[\left[\begin{matrix}5/4 & -1 \\ -1 & 1\end{matrix}\right]\]